Besicovitch's 1/2 problem and linear programming
| Autor: | De Lellis, Camillo, Glaudo, Federico, Massaccesi, Annalisa, Vittone, Davide |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We consider the following classical conjecture of Besicovitch: a $1$-dimensional Borel set in the plane with finite Hausdorff $1$-dimensional measure $\mathcal{H}^1$ which has lower density strictly larger than $\frac{1}{2}$ almost everywhere must be countably rectifiable. We improve the best known bound, due to Preiss and Ti\v{s}er, showing that the statement is indeed true if $\frac{1}{2}$ is replaced by $\frac{7}{10}$ (in fact we improve the Preiss-Ti\v{s}er bound even for the corresponding statement in general metric spaces). More importantly, we propose a family of variational problems to produce the latter and many other similar bounds and we study several properties of them, paving the way for further improvements. Comment: (changes: Added reference to A. Schechter; strengthened section 9.2) 43 pages + appendix, 10 figures. Comments are welcome |
| Databáze: | arXiv |
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